3.28.91 \(\int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx\)

Optimal. Leaf size=269 \[ -b \text {RootSum}\left [-\text {$\#$1}^8+4 \text {$\#$1}^6 d+2 \text {$\#$1}^4 c-6 \text {$\#$1}^4 d^2-4 \text {$\#$1}^2 c d+4 \text {$\#$1}^2 d^3+b-c^2+2 c d^2-d^4\& ,\frac {d \log \left (\sqrt {\sqrt {\sqrt {a x+b}+c}+d}-\text {$\#$1}\right )-\text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {a x+b}+c}+d}-\text {$\#$1}\right )}{\text {$\#$1}^5-2 \text {$\#$1}^3 d-\text {$\#$1} c+\text {$\#$1} d^2}\& \right ]+\frac {8}{15} \left (3 c+8 d^2\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}-\frac {32}{15} d \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\frac {8}{5} \sqrt {a x+b} \sqrt {\sqrt {\sqrt {a x+b}+c}+d} \]

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Rubi [B]  time = 5.08, antiderivative size = 790, normalized size of antiderivative = 2.94, number of steps used = 26, number of rules used = 13, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {1593, 6740, 194, 6688, 12, 1988, 1093, 204, 206, 1094, 634, 618, 628} \begin {gather*} 8 d^2 \sqrt {\sqrt {\sqrt {a x+b}+c}+d}-\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \log \left (-\sqrt {2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\sqrt {\sqrt {a x+b}+c}+\sqrt {\sqrt {b}-c+d^2}+d\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}+\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \log \left (\sqrt {2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\sqrt {\sqrt {a x+b}+c}+\sqrt {\sqrt {b}-c+d^2}+d\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}+\frac {\sqrt {2} \sqrt {b} \left (\sqrt {b}-c\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {b}-c+d^2}+d}-\sqrt {2} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {\sqrt {2} \sqrt {b} \left (\sqrt {b}-c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}+\frac {8}{5} \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{5/2}-\frac {16}{3} d \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{3/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tan ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

8*d^2*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - (16*d*(d + Sqrt[c + Sqrt[b + a*x]])^(3/2))/3 + (8*(d + Sqrt[c + Sqrt
[b + a*x]])^(5/2))/5 - (2*Sqrt[b]*Sqrt[Sqrt[b] + c]*ArcTan[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]/Sqrt[Sqrt[Sqrt[b]
+ c] - d]])/Sqrt[Sqrt[Sqrt[b] + c] - d] - (2*Sqrt[b]*Sqrt[Sqrt[b] + c]*ArcTanh[Sqrt[d + Sqrt[c + Sqrt[b + a*x
]]]/Sqrt[Sqrt[Sqrt[b] + c] + d]])/Sqrt[Sqrt[Sqrt[b] + c] + d] + (Sqrt[2]*Sqrt[b]*(Sqrt[b] - c)*ArcTanh[(Sqrt[d
+ Sqrt[Sqrt[b] - c + d^2]] - Sqrt[2]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]])/(
Sqrt[Sqrt[b] - c + d^2]*Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]) - (Sqrt[2]*Sqrt[b]*(Sqrt[b] - c)*ArcTanh[(Sqrt[d +
Sqrt[Sqrt[b] - c + d^2]] + Sqrt[2]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]])/(Sqr
t[Sqrt[b] - c + d^2]*Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]) - (Sqrt[b]*(Sqrt[b] - c)*Log[d + Sqrt[Sqrt[b] - c + d^
2] + Sqrt[c + Sqrt[b + a*x]] - Sqrt[2]*Sqrt[d + Sqrt[Sqrt[b] - c + d^2]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]])/(
Sqrt[2]*Sqrt[Sqrt[b] - c + d^2]*Sqrt[d + Sqrt[Sqrt[b] - c + d^2]]) + (Sqrt[b]*(Sqrt[b] - c)*Log[d + Sqrt[Sqrt[
b] - c + d^2] + Sqrt[c + Sqrt[b + a*x]] + Sqrt[2]*Sqrt[d + Sqrt[Sqrt[b] - c + d^2]]*Sqrt[d + Sqrt[c + Sqrt[b +
a*x]]]])/(Sqrt[2]*Sqrt[Sqrt[b] - c + d^2]*Sqrt[d + Sqrt[Sqrt[b] - c + d^2]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1988

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && TrinomialQ[u, x] &&  !TrinomialMatch
Q[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6740

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+x}}{\left (-b+x^2\right ) \sqrt {d+\sqrt {c+x}}} \, dx,x,\sqrt {b+a x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {\left (-c x+x^3\right )^2}{\sqrt {d+x} \left (-b+\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x^2 \left (-c+x^2\right )^2}{\sqrt {d+x} \left (-b+\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=8 \operatorname {Subst}\left (\int \frac {\left (d-x^2\right )^2 \left (c-\left (d-x^2\right )^2\right )^2}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (\left (d-x^2\right )^2+\frac {b c-b \left (c-\left (d-x^2\right )^2\right )}{-b+\left (c-\left (d-x^2\right )^2\right )^2}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (d-x^2\right )^2 \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+8 \operatorname {Subst}\left (\int \frac {b c-b \left (c-\left (d-x^2\right )^2\right )}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (d^2-2 d x^2+x^4\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+8 \operatorname {Subst}\left (\int \frac {b \left (d-x^2\right )^2}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+(8 b) \operatorname {Subst}\left (\int \frac {\left (d-x^2\right )^2}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+(8 b) \operatorname {Subst}\left (\int \left (-\frac {b+\sqrt {b} c}{2 b \left (\sqrt {b}+c-\left (d-x^2\right )^2\right )}-\frac {-b+\sqrt {b} c}{2 b \left (\sqrt {b}-c+\left (d-x^2\right )^2\right )}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+\left (4 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-c+\left (d-x^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )-\left (4 b \left (1+\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+c-\left (d-x^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+\left (4 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-c+d^2-2 d x^2+x^4} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )-\left (4 b \left (1+\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+c-d^2+2 d x^2-x^4} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+\left (2 \sqrt {b} \sqrt {\sqrt {b}+c}\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {\sqrt {b}+c}+d-x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )-\left (2 \sqrt {b} \sqrt {\sqrt {b}+c}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {b}+c}+d-x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+\frac {\left (\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}-x}{\sqrt {\sqrt {b}-c+d^2}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {\left (\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+x}{\sqrt {\sqrt {b}-c+d^2}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tan ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}}+\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {b}-c+d^2}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}+\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {b}-c+d^2}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}-\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 x}{\sqrt {\sqrt {b}-c+d^2}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 x}{\sqrt {\sqrt {b}-c+d^2}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tan ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}}-\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}-\frac {\left (2 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\sqrt {\sqrt {b}-c+d^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}-\frac {\left (2 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\sqrt {\sqrt {b}-c+d^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tan ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}}+\frac {\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {\sqrt {b}-c+d^2}}-\sqrt {2} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+\sqrt {2} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}\\ \end {align*}

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Mathematica [F]  time = 1.83, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

Integrate[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]), x]

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IntegrateAlgebraic [A]  time = 0.00, size = 246, normalized size = 0.91 \begin {gather*} -\frac {32}{15} d \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}+\frac {8}{15} \left (3 c+8 d^2+3 \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-b \text {RootSum}\left [b-c^2+2 c d^2-d^4-4 c d \text {$\#$1}^2+4 d^3 \text {$\#$1}^2+2 c \text {$\#$1}^4-6 d^2 \text {$\#$1}^4+4 d \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {d \log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )-\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c \text {$\#$1}+d^2 \text {$\#$1}-2 d \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

(-32*d*Sqrt[c + Sqrt[b + a*x]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/15 + (8*(3*c + 8*d^2 + 3*Sqrt[b + a*x])*Sqrt
[d + Sqrt[c + Sqrt[b + a*x]]])/15 - b*RootSum[b - c^2 + 2*c*d^2 - d^4 - 4*c*d*#1^2 + 4*d^3*#1^2 + 2*c*#1^4 - 6
*d^2*#1^4 + 4*d*#1^6 - #1^8 & , (d*Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1] - Log[Sqrt[d + Sqrt[c + Sqrt[b
+ a*x]]] - #1]*#1^2)/(-(c*#1) + d^2*#1 - 2*d*#1^3 + #1^5) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 57.91, size = 1404, normalized size = 5.22

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

8/5*(d + sqrt(c + sqrt(a*x + b)))^(5/2) - 16/3*(d + sqrt(c + sqrt(a*x + b)))^(3/2)*d + 8*sqrt(d + sqrt(c + sqr
t(a*x + b)))*d^2 - (b*(d + sqrt(c + sqrt(b)))^2 - 2*b*(d + sqrt(c + sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c +
sqrt(a*x + b))) + sqrt(d + sqrt(c + sqrt(b))))/((d + sqrt(c + sqrt(b)))^(7/2) - 3*(d + sqrt(c + sqrt(b)))^(5/
2)*d + 3*(d + sqrt(c + sqrt(b)))^(3/2)*d^2 - sqrt(d + sqrt(c + sqrt(b)))*d^3 - c*(d + sqrt(c + sqrt(b)))^(3/2)
+ c*sqrt(d + sqrt(c + sqrt(b)))*d) + (b*(d + sqrt(c + sqrt(b)))^2 - 2*b*(d + sqrt(c + sqrt(b)))*d + b*d^2)*lo
g(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt(d + sqrt(c + sqrt(b))))/((d + sqrt(c + sqrt(b)))^(7/2) - 3*(d + sqr
t(c + sqrt(b)))^(5/2)*d + 3*(d + sqrt(c + sqrt(b)))^(3/2)*d^2 - sqrt(d + sqrt(c + sqrt(b)))*d^3 - c*(d + sqrt(
c + sqrt(b)))^(3/2) + c*sqrt(d + sqrt(c + sqrt(b)))*d) - (b*(d - sqrt(c + sqrt(b)))^2 - 2*b*(d - sqrt(c + sqrt
(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) + sqrt(d - sqrt(c + sqrt(b))))/((d - sqrt(c + sqrt(b)))
^(7/2) - 3*(d - sqrt(c + sqrt(b)))^(5/2)*d + 3*(d - sqrt(c + sqrt(b)))^(3/2)*d^2 - sqrt(d - sqrt(c + sqrt(b)))
*d^3 - c*(d - sqrt(c + sqrt(b)))^(3/2) + c*sqrt(d - sqrt(c + sqrt(b)))*d) + (b*(d - sqrt(c + sqrt(b)))^2 - 2*b
*(d - sqrt(c + sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt(d - sqrt(c + sqrt(b))))/((d -
sqrt(c + sqrt(b)))^(7/2) - 3*(d - sqrt(c + sqrt(b)))^(5/2)*d + 3*(d - sqrt(c + sqrt(b)))^(3/2)*d^2 - sqrt(d -
sqrt(c + sqrt(b)))*d^3 - c*(d - sqrt(c + sqrt(b)))^(3/2) + c*sqrt(d - sqrt(c + sqrt(b)))*d) - (b*(d + sqrt(c
- sqrt(b)))^2 - 2*b*(d + sqrt(c - sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) + sqrt(d + sqrt(c
- sqrt(b))))/((d + sqrt(c - sqrt(b)))^(7/2) - 3*(d + sqrt(c - sqrt(b)))^(5/2)*d + 3*(d + sqrt(c - sqrt(b)))^(
3/2)*d^2 - sqrt(d + sqrt(c - sqrt(b)))*d^3 - c*(d + sqrt(c - sqrt(b)))^(3/2) + c*sqrt(d + sqrt(c - sqrt(b)))*d
) + (b*(d + sqrt(c - sqrt(b)))^2 - 2*b*(d + sqrt(c - sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + sqrt(a*x + b))
) - sqrt(d + sqrt(c - sqrt(b))))/((d + sqrt(c - sqrt(b)))^(7/2) - 3*(d + sqrt(c - sqrt(b)))^(5/2)*d + 3*(d + s
qrt(c - sqrt(b)))^(3/2)*d^2 - sqrt(d + sqrt(c - sqrt(b)))*d^3 - c*(d + sqrt(c - sqrt(b)))^(3/2) + c*sqrt(d + s
qrt(c - sqrt(b)))*d) - (b*(d - sqrt(c - sqrt(b)))^2 - 2*b*(d - sqrt(c - sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt
(c + sqrt(a*x + b))) + sqrt(d - sqrt(c - sqrt(b))))/((d - sqrt(c - sqrt(b)))^(7/2) - 3*(d - sqrt(c - sqrt(b)))
^(5/2)*d + 3*(d - sqrt(c - sqrt(b)))^(3/2)*d^2 - sqrt(d - sqrt(c - sqrt(b)))*d^3 - c*(d - sqrt(c - sqrt(b)))^(
3/2) + c*sqrt(d - sqrt(c - sqrt(b)))*d) + (b*(d - sqrt(c - sqrt(b)))^2 - 2*b*(d - sqrt(c - sqrt(b)))*d + b*d^2
)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt(d - sqrt(c - sqrt(b))))/((d - sqrt(c - sqrt(b)))^(7/2) - 3*(d -
sqrt(c - sqrt(b)))^(5/2)*d + 3*(d - sqrt(c - sqrt(b)))^(3/2)*d^2 - sqrt(d - sqrt(c - sqrt(b)))*d^3 - c*(d - s
qrt(c - sqrt(b)))^(3/2) + c*sqrt(d - sqrt(c - sqrt(b)))*d)

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maple [B]  time = 0.53, size = 190, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {16 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+8 d^{2} \sqrt {d +\sqrt {c +\sqrt {a x +b}}}+b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 d \,\textit {\_Z}^{6}+\left (6 d^{2}-2 c \right ) \textit {\_Z}^{4}+\left (-4 d^{3}+4 c d \right ) \textit {\_Z}^{2}+d^{4}-2 c \,d^{2}+c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2} d -d^{2}\right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5} d -3 \textit {\_R}^{3} d^{2}+\textit {\_R}^{3} c +\textit {\_R} \,d^{3}-\textit {\_R} c d}\right )\)	\(190\)
default	\(\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {16 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+8 d^{2} \sqrt {d +\sqrt {c +\sqrt {a x +b}}}+b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 d \,\textit {\_Z}^{6}+\left (6 d^{2}-2 c \right ) \textit {\_Z}^{4}+\left (-4 d^{3}+4 c d \right ) \textit {\_Z}^{2}+d^{4}-2 c \,d^{2}+c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2} d -d^{2}\right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5} d -3 \textit {\_R}^{3} d^{2}+\textit {\_R}^{3} c +\textit {\_R} \,d^{3}-\textit {\_R} c d}\right )\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

8/5*(d+(c+(a*x+b)^(1/2))^(1/2))^(5/2)-16/3*d*(d+(c+(a*x+b)^(1/2))^(1/2))^(3/2)+8*d^2*(d+(c+(a*x+b)^(1/2))^(1/2
))^(1/2)+b*sum((-_R^4+2*_R^2*d-d^2)/(-_R^7+3*_R^5*d-3*_R^3*d^2+_R^3*c+_R*d^3-_R*c*d)*ln((d+(c+(a*x+b)^(1/2))^(
1/2))^(1/2)-_R),_R=RootOf(_Z^8-4*d*_Z^6+(6*d^2-2*c)*_Z^4+(-4*d^3+4*c*d)*_Z^2+d^4-2*c*d^2+c^2-b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}}{\sqrt {d + \sqrt {c + \sqrt {a x + b}}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b)*sqrt(c + sqrt(a*x + b))/(sqrt(d + sqrt(c + sqrt(a*x + b)))*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}}{x\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c + (b + a*x)^(1/2))^(1/2)*(b + a*x)^(1/2))/(x*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)),x)

[Out]

int(((c + (b + a*x)^(1/2))^(1/2)*(b + a*x)^(1/2))/(x*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)**(1/2)*(c+(a*x+b)**(1/2))**(1/2)/x/(d+(c+(a*x+b)**(1/2))**(1/2))**(1/2),x)

[Out]

Timed out

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